Editing Chen's theorem (section)

== Variations ==
Chen's 1973 paper stated two results with nearly identical proofs.<ref name="Chen 1973" />{{Rp|158}} His Theorem I, on the Goldbach conjecture, was stated above. His Theorem II is a result on the [[Twin prime|twin prime conjecture]]. It states that if ''h'' is a positive even [[integer]], there are infinitely many primes ''p'' such that ''p''&thinsp;+&thinsp;''h'' is either prime or the product of two primes.

Ying Chun Cai proved the following in 2002:<ref>{{cite journal | last=Cai | first=Y.C. | title=Chen's Theorem with Small Primes| journal=Acta Mathematica Sinica | volume=18 | year=2002 | pages=597–604 | doi=10.1007/s101140200168 | issue=3| s2cid=121177443 }}</ref>

{{bi|left=1.6|''There exists a natural number <math>N</math> such that every even integer <math>n</math> larger than <math>N</math> is a sum of a prime less than or equal to <math>n^{0.95}</math> and a number with at most two prime factors.''}}

In 2025, Daniel R. Johnston, Matteo Bordignon, and Valeriia Starichkova provided an explicit version of Chen's theorem:<ref>{{cite arXiv|first1=Daniel R.|last1=Johnston|first2=Matteo|last2=Bordignon|first3=Valeriia|last3=Starichkova|eprint=2207.09452 |title=An explicit version of Chen's theorem |class=math.NT |date=2025-01-28}}</ref>
{{bi|left=1.6|''Every even number greater than <math>e^{e^{32.7}} \approx 1.4 \cdot 10^{69057979807814}</math> can be represented as the sum of a prime and a square-free number with at most two prime factors.''}}
which refined upon an earlier result by Tomohiro Yamada.<ref>{{cite arXiv|last=Yamada |first=Tomohiro |eprint=1511.03409 |title=Explicit Chen's theorem |class=math.NT |date=2015-11-11}}</ref>
Also in 2024, Bordignon and Starichkova<ref>{{Cite journal |first1=Matteo |last1=Bordignon |first2=Valeriia |last2=Starichkova |title=An explicit version of Chen’s theorem assuming the Generalized Riemann Hypothesis|date=2024 | journal=The Ramanujan Journal | doi=10.1007/s11139-024-00866-x | volume=64 | pages=1213–1242|arxiv=2211.08844 }}</ref> showed that the bound can be lowered to <math>e^{e^{14}} \approx 2.5\cdot10^{522284}</math> assuming the [[Generalized Riemann hypothesis]] (GRH) for [[Dirichlet L-function]]s.

In 2019, Huixi Li gave a version of Chen's theorem for odd numbers. In particular, Li proved that every sufficiently large odd integer <math>N</math> can be represented as<ref>{{cite journal | last=Li | first=H. | title=On the representation of a large integer as the sum of a prime and a square-free number with at most three prime divisors| journal=Ramanujan J. | volume=49 | year=2019 | pages=141–158 }}</ref>

: <math> N=p+2a, </math>

where <math>p</math> is prime and <math>a</math> has at most 2 prime factors. Here, the factor of 2 is necessitated since every prime (except for 2) is odd, causing <math> N-p </math> to be even. Li's result can be viewed as an approximation to [[Lemoine's conjecture]].